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2Lots of students are taught Trigonometry in math classLots of students are taught Trigonometry in math class. The students learn it by memorizing the concepts. Hopefully they can grasp the concept well enough and can use it to solve problems. There is sine and cosine, Cartesian coordinates, angles and right sides that everybody learns about.Education classes do not teach students how these concepts apply in the real world often enough. This presentation will show how Trigonometry is applied to making things. It will give an overview of how a machinist uses trig to make a part with a milling machine.

3The Milling MachineThis is a milling machine. It is a machine tool used to drill and bore holes, cut slots, and create flat surfaces on material. If you are familiar with a drill press, you can think of a milling machine as a heavy duty drill press that has a table that can move in three directions. Work gets mounted to the table and the cutter mounts in the spindle. You move the table with the workpiece under the spinning cutter, and cut away material.That is basically what a milling machine (mill) is used for. In the hands of a skilled machinist, a mill can do much more, such as make angled surfaces, produce elliptical surfaces, and even make gears. A mill can even generate a sphere!

4DROYXFor this example of using Trigonometry on the milling machine, we will concentrate on the table movement, or the mills ability to locate holes. Shown here are the X and Y directions of table travel. The X and Y values for table position are displayed on the digital readout or DRO. These values relate directly to the Cartesian coordinate system.

5Here is a blueprint of a flange containing six bolt holesHere is a blueprint of a flange containing six bolt holes. This is typically all the information that the engineer gives to the machinist to make a part. Notice that only one hole location is given, and all the others have to be calculated or inferred.Given is only the bolt hole circle radius.The machinist needs to use Trigonometry to calculate these hole locations.RadiusThe next series of slides will show how these hole locations are calculated using Trigonometry.

6Notice that all hole dimensions will be off the center of the bolt circle, or X 0, Y 0.

71For the first hole, we see that the X value is zero and the Y value is the radius. They are both in a positive quadrant.The first hole is at location:X 0Y 1.000

91For the third hole, we see that the X and Y values are also the same as hole two. The X value is in a positive quadrant and the Y value is in a negative quadrant.3The third hole is at location:XY

101For the fourth hole, we see that the X value is zero and the Y value is the radius, but it is a negative quadrant.4The fourth hole is at location:XY –1.000

111For the fifth hole, we see that the X and Y values are also the same as hole two. The X and Y values are in the negative quadrant..5The fifth hole is at location:X –0.866Y

121For the sixth hole, we see that the X and Y values are also the same as hole two. The X value is in a negative quadrant and the Y value is in a positive quadrant.6The sixth hole is at location:X –0.866Y

13Check marks indicate entered data.A useful tool to help students visualize how the Sine and Cosine functions relate to Right-Triangle Trigonometry is Plane Triangle Solver.In this online application, you have to enter the A) 60 degrees between holes, B) 90 degrees of the right-triangle and b) 1 for the one inch radius. After entering these three variables the program calculates all the attributes of the triangle. Notice that it solves for the X value of and the Y value of The key to using this application is knowing how to enter the variables to get the correct answer. The three-colored triangles shown above represent how pieces of known data were entered into the solver program.

15Here is another blueprint of a flange, this time containing five bolt holes. This is a typical part drawing. Notice again that only one hole location is given, and all the others have to be calculated or inferred.Given is only the bolt hole circle radius.The machinist needs to use Trigonometry to calculate these hole locations.RadiusThe next series of slides will show how these hole locations are calculated using Trigonometry.

16Notice that all hole dimensions will be off the center of the bolt circle, or X 0, Y 0.

171For the first hole, we see that the X value is zero and the Y value is the radius. They are both in a positive quadrant.The first hole is at location:X 0Y 1.000

22Another useful tool to help students visualize how the Sine and Cosine functions relate to Right-Triangle Trigonometry is Trigonometry Realms.In this online application, for hole #2 you enter the D) 72 degrees between holes and f) 1 for the one inch radius. After entering these two variables the program calculates all the attributes of the triangle. Notice that it solves for the X value of and the Y value of For Hole #3 is referencing off the 180* plane, so use 180 – 144 = 36. Enter 36 degrees for the location of hole #3. The program will then calculates X and Y Be sure to obey the signs for each quadrant and plug the calculated values into holes #4 and #5.